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Unformatted text preview: 11 Lorenz equations In this lecture we derive the Lorenz equations, and study their behavior. The equations were first derived by writing a severe, loworder truncation of the equations of RB convection. One motivation was to demonstrate SIC for weather systems, and thus point out the impossibility of accurate longrange predictions. Our derivation emphasizes a simple physical setting to which the Lorenz equations apply, rather than the mathematics of the loworder truncation. See Strogatz, Ch. 9, for a slightly different view. This lecture derives from Tritton, Physical Fluid Dynamics, 2nd ed. The derivation is originally due to Malkus and Howard. 11.1 Physical problem and parameterization We consider convection in a vertical loop or torus, i.e., an empty circular tube: cold hot g We expect the following possible ows: Stable pure conduction (no uid motion) Steady circulation Instabilities (unsteady circulation) 109 The precise setup of the loop: T T 1 (external) (T +T 3 T +T 2 0 T +T 1 (external) z (T T 3 ) ) T T 2 q a g = position round the loop. External temperature T E varies linearly with height: T E = T T 1 z/a = T + T 1 cos (24) Let a be the radius of the loop. Assume that the tubes inner radius is much smaller than a . Quantities inside the tube are averaged crosssectionally: velocity = q = q ( , t ) temperature = T = T ( , t ) (inside the loop) As in the RayleighB enard problem, we employ the Boussinesq approximation (here, roughly like incompressiblity) and therefore assume = 0 . t Thus mass conservation, which would give u in the full problem, here gives q = 0 . (25) 110 Thus motions inside the loop are equivalent to a kind of solidbody rotation, such that q = q ( t ) . The temperature T ( ) could in reality vary with much complexity. Here we assume it depends on only two parameters, T 2 and T 3 , such that T T = T 2 cos + T 3 sin . (26) Thus the temperature difference is 2 T 2 between the top and bottom, and 2 T 3 between sides at midheight. T 2 and T 3 vary with time: T 2 = T 2 ( t ) , T 3 = T 3 ( t ) 11.2 Equations of motion 11.2.1 Momentum equation Recall the NavierStokes equation for convection: u 1 + u g T + 2 u t u = p We write the equivalent equation for the loop as q 1 p + g ( T T ) sin q. (27) = t a The terms have the following interpretation: u q u u since q/ = 0. 111 p 1 p by transformation to polar coordinates. a A factor of sin modifies the buoyancy force F = g ( T T ) to obtain the tangential component: F Fsin The sign is chosen so that hot uid rises....
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This note was uploaded on 02/10/2012 for the course MATHEMATIC 487 taught by Professor Johnopera during the Spring '11 term at Cleveland State.
 Spring '11
 Johnopera
 Equations

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